### Abstract

We introduce two algebraic propositional proof systems ℱ-script N sign script S sign and ℱ-script P sign script C sign. The main difference of our systems from (customary) Nullstellensatz and polynomial calculus is that the polynomials are represented as arbitrary formulas (rather than sums of monomials). Short proofs of Tseitin's tautologies in the constant-depth version of ℱ-script N sign script S sign provide an exponential separation between this system and Polynomial Calculus. We prove that ℱ-script N sign script S sign (and hence ℱ-script P sign script C sign) polynomially simulates Frege systems, and that the constant-depth version of ℱ-script P sign script C sign over finite field polynomially simulates constant-depth Frege systems with modular counting. We also present a short constant-depth ℱ-script P sign script C sign (in fact, ℱ-script N sign script S sign) proof of the propositional pigeon-hole principle. Finally, we introduce several extensions of our systems and pose numerous open questions.

Original language | English |
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Pages (from-to) | 83-102 |

Number of pages | 20 |

Journal | Theoretical Computer Science |

Volume | 303 |

Issue number | 1 |

DOIs | |

Publication status | Published - 28 Jun 2003 |

Event | Logic and Complexity in Computer Science - Creteil Duration: 3 Sep 2001 → 5 Sep 2001 |

### Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Theoretical Computer Science*,

*303*(1), 83-102. https://doi.org/10.1016/S0304-3975(02)00446-2